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Why do the names of many Greek letters end in “a”?

In ancient Greek, words were constrained to end in a vowel, /n/ or /s/.The Phoenecian letter names did not fit that pattern, so they were adapted to end in vowels.

Of the available vowels, nouns most frequently ended in alpha (neuters or feminines) or eta (feminines). Omega, iota and upsilon were rare noun endings; omicron was not a noun ending in ancient Greek; epsilon was restricted to the dual third conjugation. So distributionally, alpha was the easiest vowel to pick. Bisyllabic Phoenecian letter names, and letters ending in stops (t, p, k, d) all end up ending in alpha. Monosyllabic letter names alternate between iota, upsilon, and -au (which is not actually a normal Greek word ending, but is at least consistent with the rest of Greek phonotactics.)

The actual alphabetic letters are indeclinable foreign words, so this is only an analogy: the alphabet letters are neuters, but don’t decline like neuters. If you google σίγματα, which would be the expected plural of σίγμα, you’ll find Modern Greek usages—and Photius saying that the plural of σίγμα is not σίγματα.

There’s also a hint from Indo-European that Greek alpha corresponded to schwa as a “default” vowel — it corresponds to the Indo-European zero-grade (absence of a vowel).

And very significantly, alphabets are recited in order, so there’s very very strong pressure for letters to sound similar. We know that people were copying out alphabets (abecedaries) for generations, getting the hang of the new invention. So once one or two letters fitting a pattern ended in alpha, all the other letters fitting the pattern did as well.

The exceptions were the monosyllabic letters, which stayed as they were, without adding an alpha: ei (epsilon is mediaeval), ou (omicron is mediaeval), u (upsilon is mediaeval), wau (digamma is later), tau, rho, pi, mu, nu, xi. The later additions stuck with the pattern of pi: phi (a counterpart of pi), chi, psi (rhyming with phi) — and ὦ (o) for omega, matching ou for omicron.